“Friends of 5” and “friends of 10” are the heart of soroban calculation. In some books they are called small friends and big friends, or complementary numbers. Once a child understands them, every addition and subtraction becomes possible — and the same number facts support mental arithmetic for life.
This guide explains the idea, gives the complete rule tables and walks through worked examples. Keep a real abacus or the virtual soroban next to you and try each step.
Why soroban needs friends of 5 and 10
Each rod has only four lower beads (worth 1) and one upper bead (worth 5). Many calculations can be done directly: if the rod shows 2, adding 1 simply means pushing up one more bead. But if the rod shows 3 and you want to add 4, only one lower bead is free. You need another way to add 4.
The trick is to replace the number with an equivalent move. Adding 4 is the same as adding 5 and taking away 1. Adding 9 is the same as adding 10 and taking away 1. The pairs that make 5 and 10 tell you which move to use.
Friends of 5 (small friends)
The friends of 5 are 1 and 4, 2 and 3. Use them when the upper bead is free but there are not enough lower beads.
| To add | Do this | To subtract | Do this |
|---|---|---|---|
| +1 | +5, then −4 | −1 | +4, then −5 |
| +2 | +5, then −3 | −2 | +3, then −5 |
| +3 | +5, then −2 | −3 | +2, then −5 |
| +4 | +5, then −1 | −4 | +1, then −5 |
Worked example, 4 + 2: four lower beads are up, so 2 cannot be added directly. Bring down the upper bead (+5), then push down three lower beads (−3). The rod shows 5 + 1 = 6.
Worked example, 6 − 3: the rod shows 5 + 1. Only one lower bead can be removed. Push up two lower beads (+2), then raise the upper bead (−5). The rod shows 3.
Friends of 10 (big friends)
The friends of 10 are 1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5. Use them when the rod cannot hold the result: you carry 1 to the rod on the left (addition) or borrow 1 from it (subtraction).
| To add | Do this | To subtract | Do this |
|---|---|---|---|
| +1 | −9, then +10 | −1 | −10, then +9 |
| +2 | −8, then +10 | −2 | −10, then +8 |
| +3 | −7, then +10 | −3 | −10, then +7 |
| +4 | −6, then +10 | −4 | −10, then +6 |
| +5 | −5, then +10 | −5 | −10, then +5 |
| +6 | −4, then +10 | −6 | −10, then +4 |
| +7 | −3, then +10 | −7 | −10, then +3 |
| +8 | −2, then +10 | −8 | −10, then +2 |
| +9 | −1, then +10 | −9 | −10, then +1 |
“+10” means one lower bead on the rod to the left; “−10” means one bead removed from that rod.
Worked example, 9 + 3: the ones rod shows 9. Take away 7 from it (the friend of 3), leaving 2, then add 1 on the tens rod. Result: 12.
Worked example, 8 + 5: take away 5 (raise the upper bead), leaving 3, then add 1 on the tens rod. Result: 13.
Worked example, 15 − 7: the ones rod shows 5, so 7 cannot be removed. Remove 1 from the tens rod (−10), then add 3 on the ones rod (the friend of 7). Result: 8.
Combined cases: when both friends are needed
Sometimes a friend-of-10 move requires a friend-of-5 move on the same rod. These are the trickiest problems for beginners, so introduce them last.
- 7 + 6: adding 6 means −4, then +10. But the rod shows 5 + 2, and you cannot remove 4 lower beads. Use the friend of 4 for subtraction: +1, then −5. The rod now shows 3. Finally add 1 on the tens rod. Result: 13.
- 13 − 6: subtracting 6 means −10, then +4. Remove 1 from the tens rod. On the ones rod (3), adding 4 needs the friend of 4: +5, then −1. The rod shows 7. Result: 7.
If your child finds these hard, go back to separate practice of each rule for a few days. The combined moves become natural when both families are automatic.
A teaching order that works
Children learn these rules best one family at a time, with plenty of repetition.
- Say the pairs aloud without the abacus: “1 and 4 make 5, 2 and 3 make 5”. Then the same for 10.
- Practise friends-of-5 additions (+4, +3, +2, +1), then subtractions.
- Practise friends-of-10 additions, starting with +9 and +8, then subtractions.
- Introduce the combined cases, such as 5 + 6, 7 + 7, 13 − 7 or 14 − 6.
- Mix all rules in short sets of 5 to 10 problems.
Our learning path introduces the rules in this order, and the exercises let you practise one rule at a time.
Practice set with answers
Friends of 5: 3 + 4, 2 + 3, 4 + 4, 7 − 3, 6 − 4, 5 − 2. Friends of 10: 9 + 7, 8 + 4, 9 + 9, 12 − 4, 15 − 8, 11 − 3. Combined: 6 + 8, 7 + 7, 13 − 7, 14 − 6.
Answers: 7, 5, 8, 4, 2, 3 — 16, 12, 18, 8, 7, 8 — 14, 14, 6, 8.
Conclusion
Friends of 5 and friends of 10 turn the soroban from a counting tool into a calculation tool. Learn the pairs that make 5 and 10, use the tables above, and practise one family at a time before mixing. With a few minutes of daily practice, these moves become automatic — and they prepare children for mental calculation.
Next: discover when and how to introduce anzan, where children apply the same rules in their heads.
Frequently asked questions
What are friends of 5 in soroban?
They are the pairs that make 5: 1 and 4, 2 and 3. For example, to add 4 when there are not enough lower beads, you add 5 and take away 1.
What are friends of 10 in soroban?
They are the pairs that make 10: 1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5. To add 9, you take away 1 and add 10 on the next rod.
Should children learn friends of 5 or friends of 10 first?
Friends of 5 first. They involve only one rod and prepare the logic used with friends of 10.
How long does it take to master friends of 5 and 10?
With short daily practice, many children are comfortable with friends of 5 in a few weeks and with friends of 10 within one to three months.
Are small friends and big friends the same as friends of 5 and 10?
Yes. “Small friends” is another name for friends of 5, and “big friends” for friends of 10.